How does the water cycle change during the year?

The water cycle technically does not change during the year.

The basic water cycle includes evaporation, condensation, and precipitation. It can get more complicated as you add more details (evapotranspiration, for example, is when water is pulled by heat energy out of plants rather than directly from a water source) but the basics remain the same.

When light energy from the sun reaches a body of water, however large or small, the atoms within each water molecule begin to gain energy and move faster, breaking and creating old and new bonds more and more quickly. The new versatility of the molecules evaporates the water- turning it into a gas, which is freer-moving and tending to rise.

The water vapor rises into the sky and cools as it does, finally condensing, or returning to a mostly liquid state as clouds.

These clouds gather more and more water until they cannot gather any more, and then they release their moisture in precipitation in the form of rain, snow, sleet, hail, or etc.

This process does not change- but the type of precipitation does change, which I believe is what you may be referring to. Depending on the temperature when precipitation happens, precipitation may occur in any one of the aforementioned forms and even several more. However, the temperature effecting the type of precipitation has much, much more to do with the position of the Earth in relation to the sun than it has to do with the actual process of the water cycle.